Zero-energy solutions and vortices in Schrodinger equations

Two-dimensional Schrodinger equations with rotationally symmetric potentials [V-a(rho)=-a(2)g(a)rho(2(a-1)) with rho=rootx(2)+y(2) and anot equal0] are shown to have zero-energy states. For the zero energy eigenvalue the equations for all a are reduced to the same equation representing two-dimensional free motions in the constant potential V-a=-g(a) in terms of the conformal mappings of zeta(a)=z(a) with z=x+iy. Namely, the zero-energy eigenstates are described by plane waves with the fixed wave numbers k(a)=root2mg(a)/(h) over bar in the mapped spaces. All the zero-energy states are infinitely degenerate similar to the case of the parabolic potential barrier (PPB) shown by Shimbori and Kobayashi [J. Phys. A 33, 7637 (2000)]. Following hydrodynamical arguments, we see that such states describe stationary flows around the origin, which are represented by the complex velocity potentials W-a=root2g(a)/mz(a), and their linear combinations create almost arbitrary vortex patterns. Examples of the vortex patterns in constant potentials and PPB are presented. In the extension to three-dimensional problems with potentials being separable into 2+1 dimensions we show that the states in three dimensions have the same structure as the two-dimensional states with the zero energy but they can generally have nonzero total energies.